It is discussed the similarities and differences of the difference equations and digital differential equations (the later have two degrees of precision while the former only one).
APPENDIX
Here are some new methodologies of solving ordinary and partial differential equations within the new axiomatic digital mathematics based on the concept of resolution which is deeper in mathematical ontology than just a grid of points in numerical methods of ODE and PDE. We apply it as example on solving the partial differential equations of Navier-Stokes equations of compressible Newtonian fluids. We deviate from our standard approach to be always only inside the new axiomatic digital mathematics, and we create new techniques within the classical mathematics with the infinite inspired by the techniques of the axiomatic digital mathematics.
Creating 3D mental images with the imagination
The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality.
Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is UTILIZED BY ME AND is required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.
Rules for phantasy and drawing of figures.
As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.
THE RESOLUTIONS METHODOLOGY IN FINDING AND STUDYING SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS (E.G. NAVIER-STOKES PARTIAL DIFFERENTIAL EQUATIONS).
1)The finite resolutions and their triple equality of quantities (ε,
δ, α) , =1, =2, =3 .
(The
definition of the finite resolution does no include the intermediate
points inside its cuboids. It is a finite set of points, a double
lattice, a double grid. But we give a separate definition for
infinite resolution when the space-time is unbounded. We prefer to
work the NsE project on the periodic formulation, on the compact
3-dimensional torus T3. We give also symbolization of the finite
resolution, based on the decimal representation of the real numbers
thus of R^n that makes it clear, and also clear the relative
distance of the unit pf space, from the minimum pixel size of minimum
invisible cuboids=pixels . The mid (double) precision cuboids of a
resolution may be already invisible, as elementary volumes, and
inside the visible points (low precision) , while the pixels are
inside the mid-precision cuboids and are the high (triple) precision.
Below we shall see that the high-precision pixels will represent the
particles of the fluid. While, the mid-precision cuboids the
non-moving elementary volumes on which we observe and state the
equations of a fluid. The differences of the values of the magnitudes
of the fluid on the adjacent mid-precision elementary volumes, are
already below the single precision of the measurement magnitude of
the fluid that is the single (low) precision. The single or low
precision defines the visible points of the fluid. The
bins (ε,
δ, α) refer to the
visibe points invisble elementary volumes and particle size. The (ε,
δ, α) is called the
signature of the resolution.There is a natura ordering of resolutions
R1<R2 if for their signatures holds
(ε1,
δ1,
α1)
<=(ε2,
δ,2
α2)
respectivelly).
2) Existence or definition of functions on finite resolutions
(Every classical function on R^n space by restriction
(inheritance) defines a function finite or infinite resolution. On
many resolutions many functions. Conversely we give conditions, that
functions on all finite or infinite resolutions as an inductive
system define a unique function of R^n or torus T3)
3)Equality of functions on finite resolutions
(Two
functions of R^n or torus T3, equal on all resolutions (infinite or
finite respectively) are equal functions on R^n or torus T3)
Bottom-up
inheritance of functions from a resolution to coarser resolution by
averaging or interpolation (Zoom-out).
Here
only the e is increased from the signature (e,d,a). That is only the visible points cuboids are increased not the elementary volumes or
particles.
Top-down
extension of a function from a resolution to a finer resolution
(Zoom-in). Here both the e and d is decreased from the
signature (e,d,a). That is the visible points cuboids and the elementary volumes are decreased in size, are refined, not the
particles.
5)The continuity on finite resolutions
(The
continuity on a resolution is defined by the pixel (infinitesimals)
of double decimal precision , as the Leibnitz continuity. No limits.
Although the ε-δ
definition of Weierstrass does apply very
well.)
6)Derivative on finite resolutions
(Similar
the definition of derivative and partial derivatives as quotients of
double decimal precision pixels (infinitesimals), like the Leibnitz
way. No limits)
7)Integration on finite resolution.
(
The integral on a finite
resolution is simply a finite sum, and double decimal precisions
pixels (infinitesimals) are involved)
8)The smoothness on finite resolution
(Smoothness is defined on finite resolution with partial
derivatives )
9) The physical natural real minimal resolution R(ε0,
δ0,
α0).
For
all physical applications of the pfuid dynamics it is required the
minimal physical resolution. Hyper-Continuity is referring to this as
well as Hyper-smoothness.
10) The Cauchy-Kovalevskaya
theorem of local existence and uniqueness of solutions and the
resolution-NSE (on a finite resolution).
Since the equality on a finite resolutions is only
single-precision on it, the resolution-NSE has
analytic coefficients, as the Taylor expansion becomes
zero after a finite number of terms. Therefore the C-K local
existence-uniqueness for quasilinear systems of PDE, applies provided
we extent the NSE that are quadratic PDE to linear PDE with
additional variables. But as we shall see later, we can do better and
derive existence and uniqueness of global solutions too.
11) Definition of functions in the
classical sense from that of an inductive system
of resolutions larger than the minimal
physical.
(We
give conditions, that functions on all finite or infinite resolutions
as an inductive system define a unique function of R^n or torus T3)
12) Characterization of equality of
functions in the classical sense from that of an
inductive system of finite resolutions larger
than the minimal physical.
(Two functions of R^n or torus T3, equal on all
resolutions (infinite or finite respectively) are equal functions on
R^n or torus T3)
13) Characterization of classical
continuity from an inductive system of
resolutions larger than the minimal physical.
Classical continuity means also continuity on the minimal physical
resolution.
(Functions
on R^n or T3, that when restricted on any resolution are continuous,
are also continuous in the classical calculus sense. The e-d of the Weierstrass apply here. If a function is continuous in the classical
sense then the for every e, there is e-1st precision resolution with d-2nd precision such that the inherited function on the resolution r is
resolution-continuous. Conversely if the function is discontinuous
some where say at a, the there is a resolution with e-1st-precison,
such that for any d-2nd-precision the inherited function is
resolution-discontinuous. NEVERTHELESS DOES NOT MEAN THAT IF WE
HAVE A DISCONTINUOUS IN THE CLASSICAL SENSE FUNCTION, THERE IS NO
RESOLUTION SUCH THAT THE INHERITIED
FUNCTION IS NOT CONTINUOUS! THERE MAY EXIST A RESOLUTION THAT THE
INHERITED FUNCTION (OF THE CLASSICAL DISCONTINUOUS FUNCTION) IS
RESOLUTION-CONTINUOUS! ).
14) Characterization of classical
smoothness with smoothness from an inductive
system of resolutions larger than the
minimal physical resolution.Classical smoothness means also
smoothness on the minimal physical resolution.
(Functions
on R^n or T3, that when restricted on any resolution are smooth ,
are also in the smooth in the classical calculus sense )
15) Integration in the classical sense from
that of an inductive system of resolutions
larger than the minimal physical resolution.
Integration is also a simple summation on the minimal physical
resolution.
(For
a Lebesgue and Riemann integrable function on R^n or T3, the
integral can be calculated, from the corresponding integrals of the
restriction functions on all resolutions)
16) The NSE on finite resolutions and
the minimal physical resolution. The resolution-NSE.
The main difficulty is that
the general methods of solutions of the NSE so far, do not involve a
systematic theoretical assessment of them, so that general properties
of the solutions can be deduced. The numerical methods suffer by that
convergence is intervened. While non-numerical methods are not
general enough to cover , the general case of the NSE. That is why,
the current methodology with the resolutions is the appropriate. We
are not concerned with convergence instead we are based on the
inductive relation of resolution-solutions and general solutions.
( On a finite resolution, the NSE from
the species of PDE, goes to the species of systems of algebraic
quadratic equations, through the pixel-deltas. We call it
resolution-NSE. The elementary volumes may be interpreted as the
single precision cuboids of the resolution where density and pressure
of the double-precision cuboids or pixels within the single
precision cuboids. In fact we find here 3 different lattices, or
grids. The finite, the mid( double)-precision pixel cuboid volumes
and the high (triple) precision cuboid volumes.
The high (triple)-precision pixels will represent the
particles of the fluid. While, the mid (double)-precision cuboids the
non-moving elementary volumes on which we observe and state the
equations of a fluid. The differences of the values of the magnitudes
of the fluid on the adjacent mid-precision elementary volumes, are
double precision magnitudes and already below the single precision of
the measurement magnitude of the fluid that is the single (low)
precision. The single or low precision defines the visible points of
the fluid.
Then the resolution-NSE are simply the ( energy and )
momentum conservation of the single-precision cuboids as non-moving
spatial observatory cuboid-windows with the double precision cuboid
volumes as particles of the fluids. In other words the
elementary-volumes are not parts of the fluid are non-moving abstract
special cuboid containers to measure quantities of the fluid the
passes through them. The pixel-cuboids of triple (high)
precision represent particles of the fluid. The density represents
number of such particles in the elementary volume. The pressure, at a
wall of the elementary volume , measures the vertical to the wall
momentum of each particle that crosses the imaginary wall. Although
the density may me constant (incompressibility) the pressure may
change, for a single elementary cuboid volume, the vertical momentum
crossed at different walls may be different. We must remember that
the quantities on the elementary volumes are averages on the
pixel-particles. The velocity at the elementary volume is the average
sample velocity of the velocities of the particles. And so is the
temperature as average norm of velocity of the particles, or the
internal energy of the fluid at the elementary cuboid volume.
Similarly for the viscosity coefficient and external body-volume
forces ad that elementary volume.
Conversely solutions of such resolution-NSE on all
resolution, with appropriate inductive compatibility conditions, give
solutions of the original NSE. Existence , uniqueness, and smoothness
are preserved. Also non-smoothness and finite blow-up times , studied
by the species of algebraic quadratic equations, are preserved after
the inductive compatibility conditions. We restrict to the
2-dimensional case so that we can compare with the 3-dimesnional
case. )
17) The energy of the resolution, in
resolution-NSE0.
We may apply ,methods of energy for the velocity
generic particle of the normal digital particle fluid, and try to
prove that it is impossible , that the velocity will converge to
infinite in a finite time, thus refuting the classical result of
finite Blow-up time. In this we must take in to indispensable account
and be based critically on the non-compressibility at least in the
form of constant density, as well as the smoothness of the pressure.
We may derive from this some regularity of the statistics. Also we
may change the definition of the normal digital particle fluid , so
as to have simple proof of the above reduction to absurd.
18) The trajectory path of an elementary volume of the resolution,
at a resolution-solution of the resolution-NSE.
19) Local in time existence of resolution-smooth solutions of the
resolution-NSE, by the corresponding results in the classical
non-resolution setting.
20) Existence in 2-dimensions of global in
time resolution-smooth solutions of the resolution-NSE , by the
corresponding Ladyzhenskaya result.
In
2-dimensions the phenomenon of vortex axial stretching and split of
the vortex to smaller and smaller vortexes and eddies cannot take
place. It is a 3-dimensional phenomenon. Still this 3-dimensional
mechanism in producing a spectrum of eddies or whirls (the smaller
the higher the spin) that traditionally is considered turbulent flow,
could still be described with smooth solutions of the NSE ! The
Kolmogorov length scale of turbulence indicates that even in
turbulence there is a un upper bound to the scale of eddies and an
upper bound to the vorticities and their speed norm and thus they do
not have the effect of the velocities converging to infinite, as time
approaches a particular moment (at the finite Blow-up time) !
Of course there are turbulent flows, that may create
singularities of very small eddies in other words point of
non-smoothness of the flow.
21) The NSE on finite resolutions and
systems of intersecting (parabolic) Quadratic equations.
(We apply here the theory of quadratic surfaces, and
their normal forms. In particular we prove that the equations of the
resolution-NSE are hyperbolic type quadratic equations. )
22) Existence of resolution-NSE
resolution-solutions on finite resolution
(Itis proved that the resolution-NSE do have solutions
given the initial boundary resolution conditions. We apply the
resolution-NSE recursively on all the points of the time/space
resolution and we get a finite system of algebraic equations that we
call resolution-solution of the resolution-NSE. They represent the
energy , mass or number of particles and momentum conservation of
finite many pixel cuboids of the resolution. The elementary single
precision volumes are simply cuboids that through their magnitudes
like density, pressure , velocity etc describe the statistical
movements of the double precision pixel cuboids or particles. And
the resolution-NSE is simply in words density
times (time-acceleration+space-acceleration) of the elemtary-volumes
equals force as pressure space-change on elementary volumes and
viscosity loss of momentum proportional to second derivatives of
the velocity.
Here the relation of Du/dt=∂u/∂t
+u(inverse∆)u is utilized (derivative rule of product and
composite)
Conservation of the mass or number of particles
(continuity equation)
All of these magnitudes as sample-group averages of the
sample of pixel-particles in the elementary-volumes of the fluid. The
initial Newton’s equation on an elementary volume is a solvable
differential equation, and so should be the NSE, that is on a lattice
of such elementary volumes. For the existence we start with
resolution of minimal number of points we verify the existence and
then we utilize induction on the points pf the finite space-time
resolution. After all it represent motion of incompressible fluids
with energy and momentum conservation. And in out formulation we
directly represent the physical atoms of the fluid (as almost
incompressible crystal fluid) and volumes of them. If anything
must be added to the formulation of the energy and momentum
conservation beyond the resolution-NSE we do formulate it, so as
derive exactly natures existence and uniqueness of motion of fluids.
Incompressibility has a nice interpretation on resolutions as slight
compressibility that is not more than double precision pixel
magnitude. We restrict to the 2-dimensional case so that we can
compare with the 3-dimesnional case. )
23) Uniqueness of solutions of
resolution-NSE on finite resolutions
(It
is proved that the resolution-NSE not only do have
resolution-solutions given the initial boundary resolution
conditions but the resolution-solution is unique. After all it
represent motion of incompressible fluids with energy and momentum
conservation. Incompressibility has a nice interpretation on
resolutions as slight compressibility that is not more than double
precision pixel magnitude. We restrict to
the 2-dimensional case so that we can compare with the 3-dimensional case. )
24)
Local smoothness of resolution-solution of the resolution-NSE on
finite resolutions (From the
previous local classical smoothness of the resolution-solution of the
resolution-NSE, we derive local resolution-smoothness of the
resolution-solutions of the resolution-NSE. Also a smooth change in
the initial conditions will result in local smooth change in the
subsequent resolution-solutions. Resolution-solutions of the
resolution-NSE evolve smoothly in time)
25) Alternative method: Digital real fluids
observable through a finite resolution
26) The resolution-NSE of a digital real fluid
27) Existence of a normal digital real fluid, for the initial
(boundary) conditions of the resolution-NSE.
Here we take a deterministic real fluid of equal size
particles at pixels of the resolution, and at each elementary cuboid
volume inherit, the velocity of the volume, while when it changes
elementary volume it experiences a deceleration according to the NSE0
formula of viscosity, while it takes also the other elementary volume
velocity of new time step of an implicit defined resolution-solution
or keeps the same as alternative rule. The later rule is according to
the Newton law and momentum conservation, the first not! But both
rules give particle-fluid satisfying the resolutions-NSE0. The former
requires to already know one, the later not, and can be used to
define one! As time goes on in discrete steps we propagate the motion
of particles as of constant velocity along their straight line linear
trajectories inside their elementary volumes. (unless they collide or
change elementary volumes).
28) The normal digital real fluid evolves
locally continuously and smoothly in time.
All
of these changes are at double precision, therefore are continuous
and smooth at the single precision. Also when particles collide the
collision is perfect elastic, and they exchange velocity, but as it
is equal as they are in the same volume, it is as if each particle
goes through the other without colliding. In case they collide, at
the walls of the elementary volumes, again, they exchange velocities,
but as the elementary volumes are adjacent again it is a change at
double precision an zero at single precision. Therefore we may
conclude that the velocities of the particles of the normal real
fluid, are continuous in time, and so is the pressure, due to the
statistical interpretation of the pressure! Finally as the new
velocities of the normal real fluid are averages of the velocities of
the particles, we conclude that the field of velocities of the normal
real fluid (not that of the particles) is continuous and smooth (1st
derivative).
29) Derivation of a resolution-solution of
the resolution-NSE, by the evolution of a digital real fluid.
Here we define the elementary volumes velocities and
wall-pressures, and densities by the statistical average quantities
of the real fluid. Then we repeat the argument as in the derivation
of the NSE0, but this time on the normal real fluid. As it is a real
fluid that its particles follow the law of motion of Newton, so it
happens with the elementary volumes, and therefore the normal real
fluid satisfies the resolution-NSE0.
We may mention here the publication of A. Muriel
http://arxiv.org/ftp/arxiv/papers/1011/1011.6630.pdf
30) Existence and uniqueness of solutions
of NSE from that on all resolutions.
(We
prove that existence and uniqueness of solutions of the
resolution-NSE on all resolutions, derive existence and uniqueness of
the solutions of NSE.)
References
W. Hurevicz
Lectures on ordinary Differential Equations
At the end of this chapter, there is
a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.
b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.
